Answer :
Sure, let's break down the solution to these questions step-by-step.
2.1 What is the gradient of the given formula for calculating the costs?
For the given cost function [tex]\( y = 2x + 1 \)[/tex], the gradient (or slope) is the coefficient of [tex]\( x \)[/tex].
- Hence, the gradient is [tex]\( 2 \)[/tex].
2.2 When drawing a graph to represent the cost function, will the graph be increasing or decreasing? Provide a reason for your answer.
A graph is increasing if its gradient (slope) is positive and decreasing if the gradient is negative.
- Since the gradient is [tex]\( 2 \)[/tex], which is positive, the graph will be increasing. This means as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases.
2.3 Verify your answer in 2.2, by sketching the cost function using the dual-intercept method. Clearly indicate your intercepts.
To sketch the graph using the dual-intercept method, we need to determine the intercepts with the axes:
- Y-intercept: The y-intercept is where the graph crosses the y-axis (i.e., [tex]\( x = 0 \)[/tex]).
- Substituting [tex]\( x = 0 \)[/tex] into the cost function [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[ y = 2(0) + 1 = 1 \][/tex]
- Thus, the y-intercept is [tex]\( (0, 1) \)[/tex].
- X-intercept: The x-intercept is where the graph crosses the x-axis (i.e., [tex]\( y = 0 \)[/tex]).
- To find the x-intercept, set [tex]\( y = 0 \)[/tex] in the cost function [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[ 0 = 2x + 1 \implies 2x = -1 \implies x = -\frac{1}{2} \][/tex]
- Thus, the x-intercept is [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex].
By plotting these intercepts, [tex]\( (0, 1) \)[/tex] and [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex], we can see that the line connects the points indicating that the graph is indeed increasing, which matches our reasoning in 2.2.
2.1 What is the gradient of the given formula for calculating the costs?
For the given cost function [tex]\( y = 2x + 1 \)[/tex], the gradient (or slope) is the coefficient of [tex]\( x \)[/tex].
- Hence, the gradient is [tex]\( 2 \)[/tex].
2.2 When drawing a graph to represent the cost function, will the graph be increasing or decreasing? Provide a reason for your answer.
A graph is increasing if its gradient (slope) is positive and decreasing if the gradient is negative.
- Since the gradient is [tex]\( 2 \)[/tex], which is positive, the graph will be increasing. This means as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases.
2.3 Verify your answer in 2.2, by sketching the cost function using the dual-intercept method. Clearly indicate your intercepts.
To sketch the graph using the dual-intercept method, we need to determine the intercepts with the axes:
- Y-intercept: The y-intercept is where the graph crosses the y-axis (i.e., [tex]\( x = 0 \)[/tex]).
- Substituting [tex]\( x = 0 \)[/tex] into the cost function [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[ y = 2(0) + 1 = 1 \][/tex]
- Thus, the y-intercept is [tex]\( (0, 1) \)[/tex].
- X-intercept: The x-intercept is where the graph crosses the x-axis (i.e., [tex]\( y = 0 \)[/tex]).
- To find the x-intercept, set [tex]\( y = 0 \)[/tex] in the cost function [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[ 0 = 2x + 1 \implies 2x = -1 \implies x = -\frac{1}{2} \][/tex]
- Thus, the x-intercept is [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex].
By plotting these intercepts, [tex]\( (0, 1) \)[/tex] and [tex]\( \left( -\frac{1}{2}, 0 \right) \)[/tex], we can see that the line connects the points indicating that the graph is indeed increasing, which matches our reasoning in 2.2.